Gadgets, approximation, and linear programming: Improved hardness results for cut and satisfiability problems

Abstract

The notion of a gadget is a central element in combinatorial reductions. Informally speaking, a gadget is a finite structure which converts a constraint of one optimization problem into constraints of a different one. Despite their central role, no uniform method has been developed to construct gadgets required for a given reduction. In fact till recently no formal definition seems to have been given. In a recent work Bellare, Goldreich, and Sudan presented a definition motivated by their work on non-approximability results. We use their definition and come up with a linear-programming based method for constructing gadgets. Using this new method we present a number of new (computer constructed) gadgets for reductions to and from MAX 3SAT, MAX 2SAT, MAX CUT, MAX DICUT, etc. The new gadgets improve hardness results for MAX CUT and MAX DICUT showing that approximating these problems to within a factor of 16/17 and 12/13 respectively are NP-hard. Interestingly, we can also use the improved reductions to present an improved approximation algorithm for MAX 3SAT which guarantees an approximation ratio of .801.

This work appeared as: Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, and David P. Williamson, “Gadgets, Approximation; and Linear Programming”, in Proceedings of the Thirty-seventh Annual Symposium on Foundations of Computer Science (FOCS), Burlington, Vermont, 1996, pp. 617–626.

Slides of this talk are available under http://www.research.ibm.com/people/w/williamson/Talks/gadgets.ps.